The number of points on the $(n-1)$-dimensional projective space $P^{n-1}(\mathbb{F}_q)$ over a finite field $\mathbb{F}_q$ is the $q$-integer $$[n]_q := \frac{q^n-1}{q-1}.$$ In analogy, the number of points on the $(n-1)$-dimensional projective space $P^{n-1}(\mathbb{F}_{un})$ over the field with one element $\mathbb{F}_{un}$ is, simply by taking $q=1$, $[n]_1=n$. Similarly, whenever the number of $\mathbb{F}_q$ points on a variety is a polynomial in $q$, we can take the $q=1$ limit just as well.

On the flip side of the coin, the $q$-integers appear in many $q$-analogs, including representation theory of quantum groups. In this context, $q$ is often a root of unity $q=e^{\frac{2\pi i}{k}}$ or more generally a complex number, and $q\rightarrow 1$ limit is understood as a "classical limit".

The connection is probably not superficial, because there are other instances where $q$ is interpreted in two different ways. For instance, according to this slide, it's a theorem of Katz that for a smooth quasi-projective variety $X$ defined over $\mathbb{Z}$, if the number of $\mathbb{F}_q$ points is a polynomial in $q$, then it is the E-polynomial of $X$, which is a specialization of the weight polynomial $WH(X;q,t)$. On the other hand, Chuang-Diaconescu-Pan related the weight polynomial to the refined Gopakumar-Vafa expansion.

So, my question is, is there any simple explanation of the apperance of $q$ in two different guises, a power of a prime and a root of unity?

  • 3
    $\begingroup$ Somehow, prime powers and roots of unity feel "dual" to each other. Perhaps this can be made precise through the fact that as locally compact abelian groups, $\mathbb{Z}$ and $\mathbb{S}^1$ are dual to each other, but don't ask me how to relate this fact to the examples that you mention... $\endgroup$ – Tom De Medts Jan 22 at 9:27

This is very not rigorous, but it's a way of thinking about this topic which I find personally helpful. Several $\mathbb F_1$ papers contains remarks about the idea going back to Weil and Iwasawa that adjoining roots of unity is analogous to base field extensions. This means that $\mathbb F_1,\mathbb F_{1^2},\dots$ shouldnt be thought of as the same "$F_{un}$", rather you should think of $\mathbb F_{1^n}$ as $\mathbb F_1$ together with $n$-th roots of unity. Notice that the number of $\mathbb F_1$ points will be the same as the number of $\mathbb F_{1^n}$ points since plugging in $q=1$ or $1^n$ gives the same thing, however things get more interesting when considering the relation between the fields rather than individually.

Just like $\mathbb F_q$ points are the Frobenius fixed points of $\mathbb F_{q^n}$, in combinatorics land $\mathbb F_1$ points should be the points of a $\mathbb F_{1^n}$ scheme fixed by a cyclic group of order $n$. Just like how you expect to count $\mathbb F_1$ points by plugging in $q=1$ formally when the point count is uniform over all finite fields, counting $\mathbb F_1$ points in a variety over $\mathbb F_{1^n}$, should have something to do with plugging in formally $q^n=1$, or in other words some primitive $n$-th root of unity. This perspective makes cyclic sieving natural in many combinatorial contexts.


One explicit connection is to look in non-describing characteristic, i.e. over an algebraically closed field $k$ of characteristic $p$ not dividing q, so $q$ can be simultaneously a root of unity and a prime power.

Let's say we are in type $A$, then the Iwahori-Hecke algebra $\mathcal{H}_q(n)$ and the $q$-Schur algebra $S(q,n)$ over $k$ arise as endomorphism algebras of certain unipotent representations of $GL_n(\mathbb{F}_q)$ and control many aspects of the unipotent representations, in fact there is an equivalence of categories between $S(q,n)$-modules and unipotent representations (for $\mathcal{H}_q(n)$-modules there are biadjoint functors, but not equivalences in general).

For example we can see that if the order of $q$ (mod p) is bigger than $n$ then the order of $GL_n(\mathbb{F}_q)$ is prime to $p$ and hence every representation is completely reducible. This corresponds to the fact in characteristic zero that the algebras $\mathcal{H}_q(n)$ and $S(q,n)$ fail to be semisimple exactly when $q$ is a root of unity of order less than $n.$

One can relate the cases of positive characteristic and characteristic zero using model theory. In particular if you let the characteristic $p$ tend to infinity while always choosing $q$ to be a primitive $d$th root of unity then the representation theory of $S(q,n)$ (or $\mathcal{H}_q(n)$) in characteristic $p$ "converges" (in a model theoretic sense) to that of $S(\zeta,n)$ in characteristic zero (where $\zeta$ is a primitive $d$th root of unity).

Hence you can prove things about Schur algebras and Iwahori-Hecke algebras at roots of unity in characteristic zero (and therefore about quantum groups) by looking at unipotent representations of $GL_n(\mathbb{F}_q)$ in non-describing characteristic. For example it's not too hard to turn the above paragraph into an actual proof of for which values of $q$ these algebras are semisimple in characteristic zero.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.